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Full text of "Mathematical And Physical Papers - Iii"

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and t~tf for t in (18*7), and integrating with respect to t'.   We thus get
To apply this result to the case of an oscillating disk, let
d9 r     r]?(ify foe the velocity of any annulus, and G the moment of
the whole force of the fluid on the disk.    Then
o   / ix/c/o \    i
r I -r  dr; o     \#/o
an(j f\ wiH be got from (188) by substituting rF(t) for /().
We find thus
Q- =  ^Tr// . pa41   Ff (t  ,) -77............(189).
Jo                 v^i
If we suppose the angular velocity of the disk to be expressed by A sin nt, where A is constant, we must put F(t) = A sin nt in (189), and we should then get after integration the same expression for Q as was obtained in Art. 8 by a much simpler process. Suppose, however, that previously to the epoch from which t is measured the disk was at rest, and that the subsequent angular velocity is expressed by At sin nt, where At is a slowly varying function of t. Then
F(t) = 0 when t < 0,        F(t) = At sin nt when t > 0. On substituting in (189) we get
At-t, cosn(t  tfj) ~,* ......(190).
Now treating At as a slowly varying parameter, we get from a formula given by Mr Airy, and obtained by the method of the variation of parameters,
J A       n
where / denotes the moment of inertia. In the expression for G we may replace At_ti under the integral sign by At outside it, because At is supposed to vary so slowly that At-tl does not much differ from At while ^ is small enough to render the integral of